L-p multipliers and their H-1-L-1 estimates on the Heisenberg group

We give a Hormander-type sufficient condition on an operator-valued function M that implies the L-p-boundedness result for the operator T-M defined by (T(M)f)over cap = M (f) over cap on the (2n + 1)-dimensional Heisenberg group H-n. Here "over cap" denotes the Fourier transform on H-n defined in terms of the Fock representations. We also show the H-1-L-1 boundedness of T-M, parallel to T(M)f parallel to(L1) <= C parallel to f parallel to(H1), for H-n under the same hypotheses of L-p-boundedness.